Defining polynomial
\(x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{19}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 19 }) }$: | $8$ |
This field is Galois over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{19}(\sqrt{2})$, $\Q_{19}(\sqrt{19})$, $\Q_{19}(\sqrt{19\cdot 2})$, 19.4.2.1, 19.4.3.1 x2, 19.4.3.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} + 18 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{4} + 19 \) $\ \in\Q_{19}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_4$ (as 8T4) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | Not computed |